I would like people to tell me under what circumstances, if ever, it can make sense to bypass a 3rd-party recruiter who has rejected you for a position and try to contact the hiring managers directly. It would be particularly relevant to hear from someone who saw this happen when they were the hiring manager, or from someone who did that as the applicant.
Below I describe a case that happened to me, just as a concrete example from which to start a risk-benefit analysis.
I found an ad for my dream job (at least seemingly) and applied for it. The company's website says they recruit only via a recruiting agency, so I applied with the assigned recruiter. She rejected me because I have never worked before with the software XYZ, which the company makes. She added the company has already hired 2 people with no XYZ skills in that same team and now wants to balance it. After doing my own research, I am prepared to make my case for why I could fit anyway, so here I am not interested in how you can overcome missing a skill in your CV.
I am also interested in when / how to bypass the 3rd-party recruiter right from the start, but not here: Those are other questions.
Update: I did not to try to go around the recruiter and a few months later she contacted me about a 2nd role that is a prerequisite for my "dream job". Now I am working at the company in this 2nd role.
PS: I added this update believing it is useful info. If it is not, please explain why in a comment. Of course, this outcome is just 1 data point: it would be interesting to hear from others who did or did not try to bypass the recruiter.